Group Theory Q: Deriving (10.80) Detail Calc

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The discussion focuses on deriving equation (10.80) from the referenced text in "Quantum Mechanics-Symmetries" by Greiner and Muller. Participants express confusion about the derivation process, suggesting that (10.80) appears to be an ansatz rather than a fully derived equation. Detailed calculations and step-by-step explanations are requested to clarify the derivation. The need for a thorough understanding of the underlying group theory principles is emphasized. Ultimately, the conversation seeks to illuminate the mathematical foundations behind (10.80) for better comprehension.
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See the attachment(from Quantum Mechanics-Symmetries 2nd W.Greiner B.Muller1994W-p345),
how to derive (10.80)
please show me the detail calculation
 

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10.80 is not derived, I think it is an ansatz
 

Thread 'Two equivalent statements of time reversal symmetric Hamiltonian'

Time reversal invariant Hamiltonians must satisfy ##[H,\Theta]=0## where ##\Theta## is time reversal operator. However, in some texts (for example see Many-body Quantum Theory in Condensed Matter Physics an introduction, HENRIK BRUUS and KARSTEN FLENSBERG, Corrected version: 14 January 2016, section 7.1.4) the time reversal invariant condition is introduced as ##H=H^*##. How these two conditions are identical?
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